Rinse the mussels thoroughly under running water. Peel and chop the onion and garlic. Clean and slice the carrots, leek, and celery into similar-sized pieces.
Heat the butter in a large saucepan over medium heat. Add the prepared vegetables and sauté for 3–4 minutes until softened.
Add the bay leaf and pour in the white wine, stirring to deglaze the pan. Bring to a boil over medium-high heat.
Add the mussels and cover with a lid. Cook for 15 minutes until the mussel shells open. Discard any mussels that remain closed after cooking.